- Meeting 01 : Mon, Jan 19, 08:00 am-08:50 am
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Administrative Announcements. Overview of the course. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. Decision vs Counting Problems. Counting the number of spanning trees in a given undirected graph. Kirchoff's Matrix Tree Theorem. The need of a theory.
- Meeting 02 : Tue, Jan 20, 12:00 pm-12:50 pm
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If #CYCLE is in FP, then P = NP. Discussion on #BPM problem. The Class #P. Closure properties.
- Meeting 03 : Tue, Jan 27, 12:00 pm-12:50 pm
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The bits and value of the #P function. Counting as a generalisation of NP, BPP, RP, PP and Parity P. NP is contained in PP.
#P = FP if and only if PP = P.
- Meeting 04 : Tue, Jan 27, 06:00 pm-06:50 pm
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(Lecture on Jan 22nd, rescheduled to Jan 27, 6pm.)
Notion of #P-completeness. Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Valiant's theorem.
- Meeting 05 : Wed, Jan 28, 06:00 pm-06:50 pm
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(Lecture on Jan 22, rescheduled to Jan 28, 6 pm)
Determinant and permanent. Combinatorial interpretation of 0-1 permanent in terms of perfect matchings. Combinatorial interpretations of permanent over integers via cycle covers in graphs with integer weights on edges. A useful trick of "cancellations" using negative weights.
- Meeting 06 : Thu, Jan 29, 11:00 am-11:50 am
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Permanent over Integers is #P-complete. Valiant's reduction.
- Meeting 07 : Fri, Jan 30, 10:00 am-10:50 am
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| Exercises | |
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Permanent over {0,1} is #P-complete.
- Meeting 08 : Mon, Feb 02, 08:00 am-08:50 am
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Viewing randomised Turing machines and complexity classes based on counting. Recal of BPP, RP, PP classes. Amplication lemma, for one-sided and two-sided error versions.
- Meeting 09 : Tue, Feb 03, 12:00 pm-12:50 pm
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| Exercises | |
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Valiant-Vazirani Lemma and its implications. Amplification. USAT vs Parity-SAT. Recap of Hash families and pairwise independence. Construction of a pairwise independent hash families.
- Meeting 10 : Thu, Feb 05, 11:00 am-11:50 am
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SAT reduces via a 1-sided randomised reduction to Parity-P with 1/n probability of success. SAT reduces via 2-sided-bounded-error randomized reduction to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplification. Plan for the rest of the proof.
- Meeting 11 : Thu, Feb 05, 06:30 pm-07:30 pm
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| Exercises | |
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Generalising the argument to show PH reduces 2-sided-error randomised reductions to Parity-P. Viewing Parity P as a modulus constraint. Modulus amplication. Plan for the rest of the proof.
- Meeting 12 : Mon, Feb 09, 08:00 am-08:50 am
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Modulus amplification. Toda's polynomials. Final P^#P Algorithm to complete Toda's theorem.
- Meeting 13 : Tue, Feb 10, 12:00 pm-12:50 pm
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BPP is contained in Sigma_2.
- Meeting 14 : Thu, Feb 12, 11:00 am-11:50 am
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Second consequence of Amplification. One random string works for all. Advice classes. P/poly. It contains all unary (even undecidable !) languages.
- Meeting 15 : Fri, Feb 13, 10:00 am-10:50 am
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To Be Announced
- Meeting 16 : Tue, Feb 17, 12:00 pm-12:50 pm
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Languages and Boolean function families. Circuit model of computation. Circuit families. Language acceptance, and parameters of circuit such as size and depth.
- Meeting 17 : Wed, Feb 18, 06:00 pm-07:00 pm
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(Compensating for Feb 18th)
PSIZE=P/poly. Uniformity in circuit complexity.
- Meeting 18 : Thu, Feb 19, 11:00 am-11:50 am
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Karp-Lipton Collapse Theorem
Meyer's Collapse Theorem
